Questions: The systolic blood pressure dataset (in the third sheet of the spreadsheet linked above) contains the systolic blood pressure and age of 30 randomly selected patients in a medical facility. What is the equation for the least square regression line where the independent or predictor variable is age and the dependent or response variable is systolic blood pressure? Patient 20 is 19 years old and has a systolic blood pressure of 124 mm Hg . What is the residual? Is the actual value above, below, or on the line? Pick What is the interpretation of the residual?

Solution Steps

The means of the independent variable \( x \) (age) and the dependent variable \( y \) (systolic blood pressure) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 40.9333 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 138.1 \]

The correlation coefficient \( r \) is computed to assess the strength of the linear relationship between \( x \) and \( y \):

\[ r = 0.9873 \]

The slope \( \beta \) of the regression line is determined using the following formulas:

Numerator for \( \beta \):

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 174560 - 30 \times 40.9333 \times 138.1 = 4973.2 \]

Denominator for \( \beta \):

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 56224 - 30 \times (40.9333)^2 = 5957.8667 \]

Thus, the slope \( \beta \) is calculated as:

\[ \beta = \frac{4973.2}{5957.8667} = 0.8347 \]

The intercept \( \alpha \) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 138.1 - 0.8347 \times 40.9333 = 103.9318 \]

The equation for the least squares regression line is:

\[ \hat{y} = 0.8347x + 103.9318 \]

For Patient 20, who is 19 years old with an actual systolic blood pressure of 124 mm Hg, we first calculate the predicted systolic blood pressure:

\[ \hat{y}_{20} = 0.8347 \times 19 + 103.9318 = 128.2089 \]

The residual is then calculated as:

\[ \text{Residual} = \text{Actual} - \text{Predicted} = 124 - 128.2089 = -4.2089 \text{ mm Hg} \]

Since the residual is negative, it indicates that the actual value is below the predicted value. Therefore, we conclude:

The actual value is below the line.

Final Answer